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 Among the following, there are two statements which can't be true together, but can be false together. Select the code that represents them. 



       Statements :

       (a) All poets are dreamers.  (b) No poets are dreamers.

       (c) Some poets are dreamers. (d) Some poets are not dreamers. <br>



 (1) (c) and (d)

  (2) (b) and (d)

  (3) (a) and (d)

  (4) (a) and (b)

This is rather confusing. because (c) and (d) are essentially the same thing isn't?

What is the right answer?

Cant get a hold on it. Can any experts clarify?

source: http://netexam.pmgurus.com/ugc-net-online-questions.aspx?q=UGC-NTA-NET-PAPER-1-solved-answer-paper-22-DECEMBER-2018-SHIFT1&gid=180&h=1&QID=12775&Qno=27

yes c) and d) are similar but not the same. The difference is that c) claims that some poets are dreamers but says nothing about no dreamers, d) is similar but for no dreamers.

Sentence a) sais that all poets are dreamer so b) and d) cannot hold if a) is true.

Sentece b) claims that all poets are no dreamers so a) and c) cannot be true if b) is true.

In predicate logic, “some $x$” means “there exists an $x$”. To see that two formulae are not equivalent, it is enough to find a structure in which one is true but the other is false.

Suppose there are only two poets and both are dreamers. Then (c) is true, while (d) is false.

Suppose that there is a box with $10$ colored balls in it (you cannot see them).

Someone takes out $2$ balls and shows them to you.

Both are green.

Now you can make the true statement "some of them are green", right?

But can you also make the statement "some of them are not green"?

Of course not: it is quite well possible that all balls in the box are green.

This indicates that the statements are definitely not the same.

edit:

Let $P$ denote the "set of poets" and $D$ the "set of dreamers". Then the statements are:

• (a) $Pcap D^{complement}=varnothing$
  


• (b) $Pcap D=varnothing$
  


• (c) $Pcap Dneqvarnothing$
  


• (d) $Pcap D^{complement}neqvarnothing$
  

(c) and (d) can both be true so option 1 falls off.

(b) and (d) can both be true so option 2 falls off.

(a) and (d) cannot both be true, but also they cannot both be false so option 3 falls off.

(a) and (b) can both be true so option 4 falls off.

So I really think that none of the options is correct.

Edit:

If you work under extra condition that $Pneqvarnothing $ (quite reasonable that poets exist) then (a) and (b) cannot both be true. They can both be false so option 4 is the correct one.

let's study the options:
option 1: c and d can be true together , because some of the poets dreamers so may be others are not . OPTION 1 IS WRONG

option 2: b and d can be true together , because when all of the poets aren't dreamers so some of them will not be also . OPTION 2 IS WRONG

option 3 : a and d can't be true together but they can't be false together. when we say not all poets are dreamers(a=false) so we mean some of them are not (d=true). OPTION 3 IS WRONG

option 4 : a and b can't be true together because we say all of poets are dreamers (a= true) so it is possible that they are not all dreamers (b= false) and the opposite is the same ( b = true ) so it is a must that (a= false)..... ALSo they can be false together . we can say that not all poets are dreamers (a= false) and at the same time not all the poets are not dreamers (b = false)> OPTION 4 IS RIGHT